Simplify (a+ square root of ab)/( square root of a+ square root of b)

$\frac{a + \sqrt{a b}}{\sqrt{a} + \sqrt{b}}$

Step 1

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a b}$ as ${(a b)}^{\frac{1}{2}}$ .

$\frac{a + {(a b)}^{\frac{1}{2}}}{\sqrt{a} + \sqrt{b}}$

Apply the product rule to $a b$ .

$\frac{a + a^{\frac{1}{2}} b^{\frac{1}{2}}}{\sqrt{a} + \sqrt{b}}$

Factor $a^{\frac{1}{2}}$ out of $a + a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

Raise $a$ to the power of $1$ .

$\frac{a^{1} + a^{\frac{1}{2}} b^{\frac{1}{2}}}{\sqrt{a} + \sqrt{b}}$

Factor $a^{\frac{1}{2}}$ out of $a^{1}$ .

$\frac{a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} b^{\frac{1}{2}}}{\sqrt{a} + \sqrt{b}}$

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} (b^{\frac{1}{2}})}{\sqrt{a} + \sqrt{b}}$

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} (b^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a} + \sqrt{b}}$

Step 2

Multiply $\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a} + \sqrt{b}} \cdot \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$

Step 3

Multiply $\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}}) (\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b}) (\sqrt{a} - \sqrt{b})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}}) (\sqrt{a} - \sqrt{b})}{{\sqrt{a}}^{2} - \sqrt{a b} + \sqrt{b a} - {\sqrt{b}}^{2}}$

Step 5

Simplify.

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}}) (\sqrt{a} - \sqrt{b})}{a - b}$